Algebraic Ideas in Ergodic Theory

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Klaus Schmidt

A co-publication of the AMS and CBMS

This book is based on a series of ten lectures sponsored by the
Conference Board on the Mathematical Sciences and presented by
the author at the University of Washington in Seattle in July 1989.
The main theme of the lectures, the influence of algebraic ideas on
the development of ergodic theory, was so extensive that the author
chose to restrict himself to two specific topics.

The first topic is the influence of operator algebras on
dynamics. The author concentrates on ergodic equivalence relations,
their properties, and their classification, presenting occasional
glimpses of the operator-algebraic context from which many of the ideas
and techniques arose. In addition, he provides a large number of
examples showing that equivalence relations provide a natural setting
for many classical constructions and classification problems.

The second topic in the book is higher dimensional Markov shifts,
a difficult field of research with no indication yet of a
satisfactory general theory. After discussing some elementary examples
of such shifts and the surprising difficulties these examples present,
the author makes the assumption that the Markov shift carries a
group structure. In that context, many of the difficulties can be
resolved, and one has the beginnings of a successful analysis which
exhibits an intriguing interplay between commutative algebra and
dynamics.

This book is based on a series of ten lectures sponsored by the
Conference Board on the Mathematical Sciences and presented by
the author at the University of Washington in Seattle in July 1989.
The main theme of the lectures, the influence of algebraic ideas on
the development of ergodic theory, was so extensive that the author
chose to restrict himself to two specific topics.

The first topic is the influence of operator algebras on
dynamics. The author concentrates on ergodic equivalence relations,
their properties, and their classification, presenting occasional
glimpses of the operator-algebraic context from which many of the ideas
and techniques arose. In addition, he provides a large number of
examples showing that equivalence relations provide a natural setting
for many classical constructions and classification problems.

The second topic in the book is higher dimensional Markov shifts,
a difficult field of research with no indication yet of a
satisfactory general theory. After discussing some elementary examples
of such shifts and the surprising difficulties these examples present,
the author makes the assumption that the Markov shift carries a
group structure. In that context, many of the difficulties can be
resolved, and one has the beginnings of a successful analysis which
exhibits an intriguing interplay between commutative algebra and
dynamics.

Book Series Name:
CBMS Regional Conference Series in Mathematics

Volume:
76

Publication Month and Year:
1990-10-09

Copyright Year:
1990

Page Count:
94

Cover Type:
Softcover

Print ISBN-13:
978-0-8218-0727-9

Online ISBN 13:
978-1-4704-2436-7

Print ISSN:
0160-7642

Online ISSN:
0160-7642

Primary MSC:
28;
54

Secondary MSC:
22;
46

Textbook?:
False

Applied Math?:
False

Electronic Media?:
False

Apparel or Gift:
False

Publisher (non-AMS):
A co-publication of the AMS and Conference Board of Mathematical Sciences